Among the dead were at least 28 Covid-19 patients on ventilators, Ali al-Bayati, a spokesman for the country's independent Human Rights Commission, a semi-official body, said on Twitter.
The cause of the fire is not known but the Iraqi Civil Defense Directorate said extinguishing work in the hospital was complete and an investigation was initiated.
In a statement, the IHCHR called on the Iraq government to take a "responsible position" following the hospital fire, saying the incident was a "crime against patients who were forced by the severity of the disease as a result of Covid-19 infection to be hospitalized".
Iraq is in the midst of a severe COVID-19 wave. Civil defence teams put out flames until the early hours of the morning.More news: Caitlyn Jenner Is Officially Running for Governor of California
Numerous victims were on respirators when the blaze at Baghdad's Ibn al-Khatib hospital started with an explosion caused by "a fault in the storage of oxygen cylinders", medical sources told AFP.
He said he and others watched helplessly as one patient struggled to breathe amid the smoke. Iraqi authorities have not released to an official casualty count.
At least two doctors at the scene confirmed they believed the oxygen cylinder had caused the flames that raged through the second floor of the hospital.
Prime Minister Mustafa al-Kadhemi suspended Health Minister Hassan al-Tamimi - who is backed by the powerful Shiite leader Moqtada Sadr - amid angry calls on social media for him to be sacked, as part of a probe that would also include the governor of Baghdad.More news: Deaths and desperation in India’s capital as COVID cases rise
The cabinet also fired the director-general of the Baghdad health department in the al-Rusafa area, where the hospital is located, and the hospital's director of engineering and maintenance, according to a statement from the Health Ministry and the prime minister's office.
United Nations envoy to Iraq, Jeannine Hennis-Plasschaert, expressed his "shock and pain".
Rather than go to overcrowded or run-down hospitals, patients who can afford it have often set up oxygen tanks for their use at home.
Iraq rolled out its vaccination campaign last month and has received almost 650,000 doses of different vaccines - the majority by donation or through the Covax scheme for low and middle income nations. Witnesses said people jumped out of windows to escape the blaze. The government is urging the public to get vaccinated, but demand has been low due to widespread mistrust of the health care system and the vaccines in particular.More news: Indonesian Navy searches for missing submarine with 53 people on board